Phylogenetic analysis aims to produce a bifurcating tree, which disregards conflicting signals and displays only those that are present in a large proportion of the data. However, any character or tree conflict in a dataset allows the exploration of support for various evolutionary hypotheses. Although data-display network approaches exist, biologists cannot easily and routinely use them to compute rooted phylogenetic networks on real datasets containing hundreds of taxa. Here, we constructed an original neighbour-net for a large dataset of Asparagales to highlight the aspects of the resulting network that will be important for interpreting phylogeny. The analyses were largely conducted with new data collected for the same loci as in previous studies, but from different species accessions and greater sampling in many cases than in published analyses. The network tree summarised the majority data pattern in the characters of plastid sequences before tree building, which largely confirmed the currently recognised phylogenetic relationships. Most conflicting signals are at the base of each group along the Asparagales backbone, which helps us to establish the expectancy and advance our understanding of some difficult taxa relationships and their phylogeny. The network method should play a greater role in phylogenetic analyses than it has in the past. To advance the understanding of evolutionary history of the largest order of monocots Asparagales, absolute diversification times were estimated for family-level clades using relaxed molecular clock analyses.